1. Binary Trees
CS-2851 Dr. Mark L. Hornick

2. Review: Doubly Linked List
Each Entry object
includes a link to the Entry object that contains the next element (or null)
And a link to the Entry object that contains the previous element (or null)
Entry
Entry
Entry
CS-2851 Dr. Mark L. Hornick

3. Binary Tree - concept
BinaryTree
A Binary Tree structure is either empty or consists of an Entry (or element),
called the Root Entry (or element)
and two distinct branches
a left subtree
and right subtree
The left and right subtrees may be empty or contain elements
a left element
and right element
Root
Left
Right
CS-2851 Dr. Mark L. Hornick

4. The Binary Tree grows…
BinaryTree
The left and right subtrees may also contain distinct branches
And each of those subtrees may contain elements
Root
Left
Right
Right Left
LeftLeft
Right Right
LeftRight
CS-2851 Dr. Mark L. Hornick

5. Binary Tree taxonomy
BinaryTree
A leaf is an element whose left and right subtrees are empty.
A node is an element containing at least one subtree
CS-2851 Dr. Mark L. Hornick

6. Depth=3
Height=4
CS-2851 Dr. Mark L. Hornick

7. More taxonomy/definitions
A binary tree is a two-tree if all non-leaf elements (nodes) have, at most, two leaves
A binary tree is full if it is a two-tree where all nodes have either 0 or 2 children.
CS-2851 Dr. Mark L. Hornick

8. More taxonomy/definitions
A binary tree is perfect if it is a two-tree with all leaves at the same depth.
A binary tree is complete if it is full to the level (height-1), and all leaves at the height of the tree are to the left
CS-2851 Dr. Mark L. Hornick

9. Can we use a Linked-List approach for implementation?
Links at least mimic the graph notation
But what about connecting the entries?
Does next and previous make sense?
Then how? (maybe left and right?)
CS-2851 Dr. Mark L. Hornick

10. Q: Is the Binary Tree a List?
A: No, because there is no definitive way to index the collection
For example, what is the 6th element of the collection?
A List implies indexing, so a Binary Tree is not a List
However, we can implement it as a special type of Collection:
a Set is a special case of a Collection which has no duplicate elements
CS-2851 Dr. Mark L. Hornick

11. There is no BinaryTree class in the JCF library
There are instead other binary-tree-based classes in the Java collections framework
TreeSet (a type of Set)
TreeMap
CS-2851 Dr. Mark L. Hornick